Question 11 Mark
If $x$ varies inversely as $y,$ then
AnswerIf $x$ varies inversely as $y,$
Solution: $xy = k ($constant$) (i)$
$x = 60$ and $y = 10 $
$xy = 60 \times 10 = 600 $
$k = 60$
when $y = 2,$ then from eq. $(i) $
$x \times 2 = k $
$2x = 600 $
$x = 300$
View full question & answer→Question 21 Mark
If $d$ varies directly as $t^2$, then we can write $dt^2= k$, where $k$ is some constant.
AnswerIf $d$ varies inversely ast $t^2$ then we can write $dt^2= k$, where $k$ is some constant. Since, two quantities $x$ and $y$ are said to be in Inverse proportion, if an increases in $x$ cause a proportional decreases in $y$ and vice-versa, in such a manner that the product of their corresponding values remains constant.
View full question & answer→Question 31 Mark
Both $x$ and $y$ are said to vary _______ with each other if for some positive number $k, xy = k.$
Answer Both $x$ and $y$ are said to vary inversely with each other, if for some positive number $k, xy = k.$
View full question & answer→Question 41 Mark
If $x$ and $y$ are in inverse proportion, then $(x + 1)$ and $(y + 1)$ are also in inverse proportion.
AnswerIf $x$ and $y$ are in inverse proportion,
then $xy = k ($constant$)$ e.g.
Let $x = 2$ and $y = 3xy = 2 \times 3 = 6.$
$xy = 2 \times 3 = 6.$
Now, $x + 1 = 2 + 1 = 3$ and $y + 1 = 3 + 1 = 4$
Then, $(x + 1)(y + 1) = 3 \times 4 = 12 $[not inverse proportion]
Hence, $(x + 1)$ and $(y + 1)$ cannot be in inverse proportion.
View full question & answer→Question 51 Mark
On increasing $a, b$ increases in such a manner that $\frac{\text{a}}{\text{b}}$ remains _____ and positive, then $a$ and $b$ are said to vary directly with each other.
AnswerOn increasing $a, b$ increases in such a manner that $\frac{\text{a}}{\text{b}}$ remains constant and positive, then $a$ and $b$ are said to vary directly with each other.
View full question & answer→Question 61 Mark
If two quantities $p$ and $q$ vary inversely with each other then _______ of their corresponding values remains constant.
AnswerIf two quantities $p$ and $q$ vary inversely with each other, then product of their corresponding values remains constant
View full question & answer→Question 71 Mark
If on increasing $a, b$ decreases in such a manner that _______ remains ________ and positive, then $a$ and $b$ are said to vary inversely with each other.
AnswerIf on increasing $a, b$ decreases in such a manner that ab remains constant and positive, then $a$ and $b$ are said to vary inversely with each other.
View full question & answer→Question 81 Mark
For fixed time period and rate of interest, the simple interest is directly proportional to the principal.
AnswerFor fixed time period $(T)$ & rate of interest $(R),$ the simple interest is directly proportional to the principal.
We know that, $\text{SI}=\frac{\text{P}\times\text{R}\times\text{T} }{100}$
$\frac{\text{SI}}{\text{p}}=\frac{\text{R}\times \text{T}}{100}$ contant (as and $T$ are constants)
Simple interest is directly proportional to the principal.
View full question & answer→Question 91 Mark
If $45$ persons can complete a work in $20$ days, then the time taken by $75$ persons will be _______ hours.
AnswerIf $45$ persons can complete a work in $20$ days, then the time taken by $75$ persons will be 288 hours. Solution: $45$ persons can complete a work in $20$ days. $1$ person can complete a work in $45 × 20$ i.e., $900$ days Similarly, $75$ persons can complete the same work in $\frac{900}{75}=12$ days $=12×24=288h$
View full question & answer→Question 101 Mark
If one angle of a triangle is kept fixed then the measure of the remaining two angles vary inversely with each other.
Answer False
Solution:
If one angle of a triangle is kept fixed, then the measure of the remaining two angles can’t vary inversely with each other.e.g.,
In $=\text{ABC}, \angle\text{A}+ \angle\text{B} + \angle\text{C} = 180^\circ$
If $\text{A}= 50^\circ$ then, $\text{B}+\text{C}=180^\circ-50^\circ=130^\circ$
So, it is not depend on change any proportion by applying angle sum properties of a triangle.
View full question & answer→Question 111 Mark
If $x$ varies inversely as $y$ and when $x = 6, y = 8,$ then for $x = 8$ the value of $y$ is $10.$
AnswerIf $x$ varies inversely as $y,$ i.e., $xy = k$ (constant) If $x = 6$ and $y = 8$
$xy = 6 \times 8 = 48$ But if $x = 8$ and $y = 10$
$xy = 8 \times 10 = 80$ Here, $ 48\neq80$
Hence, the value of $y$ is not $10.$
View full question & answer→Question 121 Mark
When two quantities are related in such a manner that, if one increases, the other also increases, then they always vary directly.
AnswerTrueSolution:
When two quantities are related in such a manner that if, one increases the other also increases, then they always vary directly. Above statement is correct for direct proportion. It is a basic properties of direct proportion.
View full question & answer→Question 131 Mark
When the speed is kept fixed, time and distance vary inversely with each other.
AnswerFalse Solution: When the speed is kept fixed, time and distance vary directly with each other.
View full question & answer→Question 141 Mark
The number of workers and the time to complete a job is a case of direct proportion.
AnswerFalse Solution: The number of workers and the time to complete a job is a case of indirect proportion, e.g. If $60$ workers can complete a work in $10$ days. Then, $120$ workers can complete the same work in 5 days.
View full question & answer→Question 151 Mark
If the thickness of a pile of $12$ cardboard sheets is $45\ mm,$ then the thickness of a pile of $240$ sheets is ________ $cm.$
AnswerThe thickness of a pile if $12$ cardboards sheet $= 45\ mm$
The thickness of a pile of $1$ cardboard sheet $\frac{45}{12}\text{mm}$
So, the thickness of a pile of $240$ cardboard sheet $=\frac{45}{12}\times240\text{mm}$
$= 45 × 20 = 900\ mm$
$=\frac{900}{10}\text{cm}=90\text{cm}$
View full question & answer→Question 161 Mark
In case of inverse proportion, $\frac{\text{a}_2}{-}=\frac{\text{b}_2}{-}.$
Answer In case of inverse proportion, $\frac{\text{a}_2}{\text{a}_1}=\frac{\text{b}_2}{\text{b}_1}.$
Solution:
$\text{a}_2\text{b}_1=\text{a}_1\text{b}_2$
View full question & answer→Question 171 Mark
If two quantities $x$ and $y$ vary directly with each other, then _______ of their corresponding values remains constant.
AnswerIf two quantities $x$ and $y$ vary directly with each other, then ratio of their corresponding values remains constant.
View full question & answer→Question 181 Mark
The area of cultivated land and the crop harvested is a case of direct proportion.
AnswerTrue Solution: The area of cultivated land and the crop harvested is a case of direct proportion. Since, the quantities of crop harvested is depend upon area of cultivated land
View full question & answer→Question 191 Mark
An auto rickshaw takes $3$ hours to cover a distance of 36km. If its speed is increased by $4\ km/h,$ the time taken by it to cover the same distance is ________ .
AnswerThen, its speed $=\frac{36}{3}=12\text{km}/\text{h}$
If its speed increases $4\ km/h ,$
then New speed$ = 12 + 4 = 16\ km/h$
Now, time taken by auto rickshaw to cover $36\ km$ in$\frac{35}{16}\text{h}=\frac{36\times60}{16}=135\text{min}$
$2 \times 60 + 15 = 2\ h\ 15\ min$
View full question & answer→Question 201 Mark
$x$ and $y$ are said to vary directly with each other if for some positive number $k,$______$ = k.$
Answer$x$ and $y$ are said to vary directly with other, if for some positive number $k,$ $\frac{\text{x}}{\text{y}}=\text{k}.$
View full question & answer→Question 211 Mark
If $x$ vartes directly as $y,$ then
Answer$\frac{\text{x}}{\text{y}}=\text{k}$ (constant)
If $x = 12$ and $y = 48,$
then $\frac{\text{x}}{\text{y}}=\frac{12}{48}=\frac{1}{4}$
$\text{k}=\frac{1}{4}$ When $x = 6$
then from eq. $(i) \frac{6}{\text{y}}=\text{k}$
$ \frac{6}{\text{y}}=\frac{1}{4}$
$6\times 4=y\times 1$
$y=24$
View full question & answer→Question 221 Mark
When the speed remains constant, the distance travelled is ________ proportional to the time.
AnswerWhen the speed remains constant, the distance travelled is directly proportional to the time.
View full question & answer→Question 231 Mark
Length of a side of an equilateral triangle and its perimeter vary inversely with each other.
AnswerLength of $a$ side of an equilateral triangle and its perimeter vary directly with each other, e.g.
Let $a$ be the side of an equilateral triangle.
So, perimeter $= 3 \times ($side$) = 3 \times a = 3a.$
So, if we increase the length of side of the equilateral triangle, then their perimeter will also increases.
View full question & answer→Question 241 Mark
If $x$ and $y$ are in direct proportion, then $(x - 1)$ and $(y - 1)$ are also in direct proportion.
AnswerIf $x$ and $y$ are in direct proportion,
then $\frac{\text{x}}{\text{y}}=\text{k}$ (constant) e.g., let $x = 4 $and $y = 6$
$\frac{\text{x}}{\text{y}}=\frac{4}{6}=\frac{2}{3}$
Now, $x - 1 = 4 - 1 = 3$ and $y - 1 = 6 - 1 = 5$
$\frac{\text {x}-1}{\text{y} -1}=\frac{3}{5}$
View full question & answer→Question 251 Mark
The height of a tree and the number of years.
AnswerThe height of a tree and the number of years are neither directly nor inversely proportional to each other.
View full question & answer→Question 261 Mark
Two quantities are said to vary ______ with each other if an increase in one causes a decrease in the other in such a manner that the product of their corresponding values remains constant.
AnswerTwo quantities are said to vary inversely with each other if an increase in one causes a decrease in the other in such a manner that the product of their corresponding values remains constant.
View full question & answer→Question 271 Mark
When two quantities $x$ and $y$ are in ______ proportion or vary _____ they are written as $\text{x}\propto\frac{1}{\text{x}}$
AnswerWhen two quantities $x$ and $y$ are in inverse proportion or vary inversely, they are written as $\text{x}\propto\frac{1}{\text{x}}$
View full question & answer→Question 281 Mark
Devangi travels $50 m$ distance in $75$ steps, then the distance travelled in $375$ steps is ________ $km.$
AnswerDevangi travels $50\ m$ distance in $75$ steps, then the distance travelled in $375$ steps is $0.25\ km.$
Devangi covers the distance in $75$ steps $= 50m$
So, she cover the distance in $1$ step $=\frac{50}{75}\times375$
$=\frac{18750}{75}=250\text{m}$
$=\frac{250}{1000}\text{km}=0.25\text{km}$
View full question & answer→Question 291 Mark
If $xy = 10,$ then $x$ and $y$ vary ______ with each other.
AnswerIf $xy = 10,$ then $x$ and $y$ vary inversely with each other.
Solution:
Given, $\text{xy}=10 $
$\text{x}=\frac{10}{\text{y}}$
$x$ and $y$ vary inversely with each other.
View full question & answer→Question 301 Mark
A car is travelling $48\ km$ in one hour. The distance travelled by the car in $12$ minutes is _________.
AnswerA car is travelling $48\ km$ in one hour. The distance travelled by the car in $12$ minutes is $=\frac{48}{60}\times12=\frac{48}{5}=9.6\text{km}$ .
Solution:
A car travelling in $1\ h = 48\ km$
So, car travelling in $1\ min =\frac{48}{60}\text{km}$
Similarly, car travelled in $12\ min =\frac{48}{60}\times12=\frac{48}{5}=9.6\text{km}$
View full question & answer→Question 311 Mark
When the distance is kept fixed, speed and time vary directly with each other.
AnswerFalse Solution: When the distance is kept fixed, speed and time vary directly with each other. Since, if we increase speed, then taken time will less & vice versa.
View full question & answer→Question 321 Mark
If a tree $24m$ high casts a shadow of $15m,$ then the height of a pole that casts a shadow of 6m under similar conditions is $9.6m.$
AnswerHeight of a tree $= 24m.$
Then, its shadow $= 15m$
Similarly condition, if a pole has a shadow of length $= 6m$
Let the height of pole $= x\ m$
Since, length and shadow vary directly. Then,
$\frac{24}{15}=\frac{\text{x}}{6} $
$15\times\text{x}=24\times6$
$\text{x}=\frac{24\times6}{15}=9.6\text{m}$
View full question & answer→Question 331 Mark
When two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely.
AnswerTrue Solution: When, two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely. Above statement is correct for inverse proportion. It is a basic properties of inverse proportion.
View full question & answer→Question 341 Mark
If $12$ pumps can empty a reservoir in $20$ hours, then time required by $45$ such pumps to empty the same reservoir is ______ hours.
Answer$12$ pumps can empty a reservoir in $20h. $
$1$ pump can empty the same reservoir in
Solution:
$\frac{240}{45}\text{h}= \frac{240\times60}{45}\text{min}$
$\frac{14400}{45}=320\text{min}$
$5\times 6 + 20=5\text{h }20\text{min}$
View full question & answer→Question 351 Mark
If $x$ varies inversely as $y$ and $x = 4$ when $y = 6,$ then when $x = 3$ the value of $y$ is _________ .
AnswerIf $x$ varies inversely as $y$ and $x = 4$ when $y = 6,$
then when $x = 3$
the value of $y$ is $\text{y}=\frac{24}{3}=8$ .
Solution:
In inverse proportion, $xy = k ($constant$)$
If $x = 4$ and $y = 6$ then $k = 4 \times 6 = 24$
Now, when $x = 3,$ then $\text{y} = \frac{\text{k}}{\text{x}}$
$\text{y}=\frac{24}{3}=8$
View full question & answer→Question 361 Mark
The perimeter of a circle and its diameter vary _______ with each other.
Answer The perimeter of a circle & its diameter vary directly with each other. Solution:
Perimeter of a circle $= 2\pi \text{r}$ Diameter of a circle $= 2 \times \text{r}$ Perimeter $= \pi \times$ Diameter View full question & answer→Question 371 Mark
If $x = 5y,$ then $x$ and $y$ vary ________with each other.
AnswerIf $x = 5y,$ then $x$ and $y$ vary directly with each other.
solution:
Given, $x = 5y$
Then,$\frac{\text{x}}{\text{y}}=\frac{5}{1}=\text{k}$(constany) $x$ and $y$ vary directly with each other.
View full question & answer→Question 381 Mark
Two quantities $x$ and $y$ are said to vary directly with each other if for some rational number $k, xy = k.$
AnswerTwo quantities $x$ and $y$ are said to vary directly with each other, if $x_y= k$. (constant)
View full question & answer→Question 391 Mark
Length of a side of a square and its area vary directly with each other.
AnswerLength of a side of a square and its area does not vary directly with each other, e.g. Let a be length of each side of a square.
So, area of the square $=$ side$^2= a^2$
So, if we increase the length of the side of a square, then their area increases but not direct.
View full question & answer→Question 401 Mark
If $p$ and $q$ are in inverse variation then $(p + 2)$ and $(q – 2)$ are also in nverse proportion.
AnswerIf $p$ and $q$ are in inverse proportion, then $xy = k ($constant$)$ e.g.
Let $p = 3$ and $q = 4$
Then, $pq = 3 \times 4 = 12$
Now, $p + 2 = 3 + 2 = 5$ and $q − 2 = 4 − 2 = 2$
$(p + 2)(q – 2) = 5 \times 2 = 10$ [not in inverse proportion]
Hence, $(p + 2)$ and $(q − 2)$ cannot be in inverse proportion
View full question & answer→Question 411 Mark
If the area occupied by $15$ postal stamps is $60\text{cm}^2$, then the area occupied by $120$ such postal stamps will be _______ .
AnswerIf the area occupied by $15$ postal stamps is $60\text{cm}^2$,
then the area occupied by $120$
such postal stamps will be $= 4 \times 120 = 480 \text{cm}^2$ .
Solution:
Area occupied by $15$ postal stamps $=60\text{cm}^2$
Area occupied by $1$ postal stamp $=\frac{60}{15}=4\text{cm}^2$
Similarly, area occupied by $120$
such postal stamps $=4\times120=480\text{cm}^2$
View full question & answer→Question 421 Mark
When two quantities $x$ and $y$ are in _____ proportion or vary ______ they are written as $\text{x} \propto \text{y}$.
AnswerWhen two quantities $x$ and $y$ are in direct proportion or vary directly, they are written as $\text{x}\propto \text{y}$.
View full question & answer→Question 431 Mark
Two quantities are said to vary _______ with each other if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant.
AnswerTwo quantities are said to vary directly with each other, if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant.
View full question & answer→Question 441 Mark
In direct proportion, $\frac{\text{a}_{1}}{\text{a}_1}$ ________ $\frac{\text{a}_{2}}{\text{a}_2}$
AnswerIn direct proportion, $\frac{\text{a}_{1}}{\text{a}_1}=\frac{\text{a}_{2}}{\text{a}_2}$ Solution: Where, $\frac{\text{a}_{1}}{\text{a}_2}=\text{k}$(constant)
View full question & answer→Question 451 Mark
If $x$ varies inversely as $y$ and $y = 60$ when $x = 1.5.$ Find $x$. when $y = 4.5.$
AnswerIf $x$ varies inversely as $y.$
$xy = k ($costant$)$
$x = 15$ and $y = 60$
$xy = 1.5 \times 60 = 90$
$k = 90$ When $y = 4.5,$ then from eq. $(i)$
$4.5 \times y = k$
$4.5 \times y = 90$
$\text{y}=\frac{90}{45}=20$
View full question & answer→